What happens when two identical one-sided objects are merged in space?

[Observeraren]

I'm not an expert in any field but i got this thought of merging two Moebius strips togheter the same way as merging one side of a square with another side of a square resulting in a rectangle.
Now the logic being that 1 side is subtracted(is it still a merge?) from Object A and object B what happens when i merge(?) two objects that are one-sided, are identical and in the same position in space to each other. How does the existens of such a object look like.

 

[micromass]

It's the Klein bottle

https://en.wikipedia.org/wiki/Klein_bottle

 

[jedishrfu]

If you instead cut a Moebius strip in half which is the inverse of your suggested operation, what do you get?